Regularizing effects for $u_{t}=\Delta \varphi (u)$

Abstract:

One expression of the fact that a nonnegative solution of the initial-value problem \[ ({\text {IVP}})\quad \left \{ {\begin {array}{*{20}{c}} {{u_t} - \Delta {u^m} = 0,} \\ {u(0,x) = {u_0}(x),} \\ \end {array} } \right .\quad \begin {array}{*{20}{c}} {t > 0,x \in {R^N},} \\ {} \\ \end {array} \] where $m > 0$, is more regular for $t > 0$ than a rough initial datum ${u_0}$ is the remarkable pointwise inequality ${u_t} = \Delta {u^m} \geqslant - (N/(N(m - 1) + 2)t)u$ obtained by Aronson and Bénilan for $t > 0$ and $m > \max ((N - 2)/N,0)$. This inequality was used by Friedman and Caffarelli in proving that solutions of (IVP) are continuous for $t > 0$. The main results of this paper generalize the Aronson-Bénilan inequality and show the extended inequality is valid for a much broader class of equations of the form ${u_t} = \Delta \varphi (u)$. In particular, the results apply to the Stefan problem which is modeled by $\varphi (r) = {(r - 1)^ + }$ and imply ${({(u - 1)^ + })_t} \geqslant - ({(u - 1)^ + } + N/2)/t$ in this case.